Question

Jim held a firehose whose spray formed a parabola that spanned 20m. The maximum height of the spray is 16m. What is the quadratic equation that models the path of the spray?

Answer 1

graph{-0.16x^2+3.2x [-4.41, 27.63, 1.96, 17.98]}

`y=-16/100x^2+16/5x`

Explanation:

Assuming Jim is standing at the point (0,0) facing to the right, we are told that the two intercepts (roots) of the parabola are at (0,0) and (20,0). Since a parabola is symmetrical, we can infer that the maximum point is in the middle of the parabola at (10,16).

Using the general form of the parabola: `ax^2+bx+c`

Product of roots = `c/a` = 0 therefore `c=0`
Sum of roots = `-b/a=20` therefore `20a+b=0`

We are given a third equation from the maximum point:
When x=10, y=16, i.e. `16=a*10^2+b*10+c`

Since `c=0`, and as above:

`10a+b=16/10`
`20a+b=0`

by subtraction: `-10a=16/10`
`a=-16/100`
therefore: `b=16/5`

Returning to our general form of the quadratic equation: `y=ax^2+bx+c` we can sub in values for a and b to find the equation to be:

`y=-16/100x^2+16/5x`